There are a few options for you to explore in this sort of situation. First, contact whoever is in charge of services at the church. After explaining how important the music and readings were to your loved one, the director may be able to give special permission to allow them for the services.
If this is not possible, you can consider playing the pieces or arranging the readings for another point in the services – at the funeral home or graveside, for instance.
You may also consider changing the venue for the funeral services. Think about how important the music and readings were for your loved one and how to best honor them. Most funeral directors will honor special requests – within reason, of course.
If you decide to change the venue to a secular establishment in order to honor your loved one more faithfully, consider having a more religious memorial at your church and go with the pieces and passages that the church approves.